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arXiv · 2506.16204

Heterotopic energy for Sobolev mappings

Antoine Detaille·Jean Van Schaftingen
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Abstract

We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and only if the mappings in the corresponding homotopy classes are homotopic on a codimension one skeleton of a triangulation of the domain. When this is the case, the heterotopic energy of a mapping is the sum of its Sobolev energy and its disparity energy, defined as the minimum energy of a bubble to pass between these homotopy classes. At the more technical level, we rely on a framework that works when the target and domain manifolds are not simply connected and there is no canonical isomorphism between homotopy groups with different basepoints.

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Antoine Detaille, Jean Van Schaftingen. 2025-06-19. Heterotopic energy for Sobolev mappings. https://arxiv.org/abs/2506.16204

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